What does the speaker mean by the words "blue fingers" in "The Vagabond"?

In this poem, the phrase “blue finger” is an example of synecdoche; the part (finger) is being used to represent the whole (the whole person). The speaker tells us that autumn is “silencing” a bird and “biting” the finger as a way of telling us that the cold is very strong. Using “autumn” to mean “cold” is another example of synecdoche: this time the whole (autumn) is being used to refer to one of its aspects (coldness). Synecdoche is a figurative technique in which a part is used to refer to the whole or the whole is used to refer to a part. I have included a link to the dictionary definition of synecdoche.
In terms of the idea that the speaker is trying to convey, the strength of this line is in its hyperbole: the cold is not literally “biting” the fingers, but this word makes the reader think of the coldness as almost like an aggressive animal. By using “biting the blue finger” the speaker can make the reader understand the severity of the cold without looking foolish for wanting to stay outside.
So by using the blue fingers, as a metaphor the speaker also romanticizes the cold. Imagine the effect if the poem instead referred to a “gangrenous frostbitten toe”; the reader would be far less sympathetic to the speaker’s desire for freedom. Using this imagery allows the speaker to convincingly continue on their quest to stay outside even when it’s dangerous for their health.
https://www.merriam-webster.com/dictionary/synecdoche

The speaker of the poem is a tramp who likes to spend his whole life wandering from place to place, out in the open air come rain, hail, or shine. He wants us to know that the change of seasons doesn't bother him at all. Spring, summer, fall, winter—whatever the season, whatever the weather, you'll always find him out and about wandering.
Come the winter time, the wanderer will still be roaming far afield. Although the winter may have silenced the birds in the trees, forcing them to migrate to warmer climes, the wanderer has no intention of going anywhere. He'll stay right where he is, even if the bitter cold of winter should bite into his blue, frosty fingers. If you've ever been outside on a cold winter's day without gloves, you'll know that your fingers can very quickly turn blue in the icy chill, and that's most probably what the speaker's referring to here.
The wanderer wants us to know that the prospect of spending all winter outside in the freezing weather doesn't concern him in the slightest, blue fingers or not.
https://www.poemhunter.com/poem/the-vagabond/